The TravelTime plugin for QGIS comes with a range of options for creating A to B routes, as well as turn-by-turn directions:
Quick Route - runs off map clicks to set the origin and destination, with minimal parameter configuration required
Routes Simple - runs off layers of points, with minimal parameter configuration required
Routes Advanced - runs off layers of points, allowing full configuration of all parameters
Watch below for a short walk-through of how to use these tools.